Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23439 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×#121 |
| prime radical | C22×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4638 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6)(7,17)(8,22)(9,15)(10,16)(11,14)(12,20)(13,25)(18,27)(19,28)(21,29)(23,35)(24,30)(26,31)(32,33)(34,36);(5,7)(6,12)(8,23)(9,19)(10,20)(11,21)(13,27)(14,28)(15,17)(16,29)(18,26)(22,32)(24,34)(25,33)(30,35)(31,36);(2,4)(5,8,9,24,11,26,10,25)(6,13,16,31,14,30,15,22)(7,18,20,34,21,23,19,33)(12,27,17,32,28,35,29,36);(6,14)(12,28)(13,30)(15,16)(17,29)(22,31)(27,35)(32,36);(5,9,11,10)(6,15,14,16)(7,19,21,20)(8,24,26,25)(12,17,28,29)(13,22,30,31)(18,33,23,34)(27,32,35,36)Polycyclic presentation:
2^9|c^2=f,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,g^a=gi,h^a=hi,c^b=ch,f^b=fi,g^b=gi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72316826, #72316840, #72316854, … |
| maximal normal subgroup of | #72316826, #72316840, #72316854, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339140);